According to this story, Paul Johnson, former Mayor of Phoenix, will attempt to qualify an initiative in 2016 to convert all Arizona elections, except presidential elections, to non-partisan elections. There will be no party labels on any ballot, except for President, under the proposal.

Great idea, but it was done a different way in Washington State.
First voters said that no state money can be used for any political party (stationary, phone calls, etc.)

Then, under the same law, it became against the law for the state to spend on partisan primaries. No new vote was required to get to the top two. The parties did not want to spend the money to run their own primaries.